Benford's Law. Theory, the General Law of Relative Quantities, and Forensic Fraud Detection Applications. Alex Ely Kossovsky.

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1 BEIJING SHANGHAI Benford's Law Theory, the General Law of Relative Quantities, and Forensic Fraud Detection Applications Alex Ely Kossovsky The City University of New York, USA World Scientific NEW JERSEY LONDON SINGAPORE HONG KONG TAIPEI CHENNAI

2 CONTENTS Benford's Law Foreword Introduction Acknowledgment v vii ix xiii Section 1: Benford's Law 1 1. Digits versus Numbers 3 2. To Find Fraud, Simply Examine Its Digits! S 3. First Leading Digits 8 4. Empirical Evidence from Real-Life Data on Digit Distribution 9 5. Physical Clues of the Digital Pattern 1S 6. Historical Background of the Two Discoverers Benford's Law The Prevalence of Benford's Law Physical Law versus Numerical Law Nature's Way of Counting Single-Issue Phenomena Case Study I: Time Between Earthquakes Data on Population Counts of Cities,Towns, Regions, and Districts Case Study II: U.S. Census Data on Population Centers Data sets on USA Population by State and by County Four Distinct Numerical Processes Leading to Benford Random Linear Combinations and Accounting Revenue Data Aggregation of Data Sets as a Prominent Cause of Benford's Law S3 18. Random Pick from a Variety of Data Sources is Logarithmic SS 19. Integral Powers of Ten The Logarithmic as Repeated Multiplications Case Study III: Exponential 0.5% Growth Series for 3,233 Periods Case Study IV: 140 Cumulative Dice Multiplications The Universality of Benford's Law True in any Scale System 72 xvii

3 xviii Contents 24. A Hidden Digital Signature within Benford's Digital Signature 74 Section 2: Forensic Digital Analysis & Fraud Detection Historical Background of the First Applications of Benford's Law Methods in Financial and Accounting Fraud Detection The Part and Type of Data Applicable to Forensic Testing Case Study V: U.S. Market Capitalization on January 1, Case Study VI: Microsoft Corporation Financial Statement Case Study VII: Total Return of Athena Guaranteed Futures Fund Establishing Direct Connection Between Digit Anamoly & Fraud Post-Test Conclusions Detecting Fraud via Digital Development Pattern The Dilemma of FTD versus LTD for Digit-Anemic Numbers 110 Section 3: Data Compliance Tests Testing Data for Conformity to Benford's Law The Z Test The chi-squaretest SSD as a Measure of Distance from the Logarithmic Saville Regression Measure Value Repetition Test The Confusion and Mistaken Applications of Summation Test Summation Test in the Context of Fraud Detection Methods in Digital Development Pattern Detection Case Study VIII: Price List of a Large Manufacturer Case Study IX: USA County Area Data Random Linear Combinations and Revenue Data Revisited Case Study X: Forensic Analysis of Revenue Data for Small Shop 192 Section 4: Conceptual and Mathematical Foundations Hybrid Data Sets Blending Several Data Types Second-Generation Distributions A Leading Digits Parable Simple Averaging Scheme as a Model for Typical Data More Complex Averaging Schemes Digital Proportions within the Number System Itself 216

4 Contents xix 54. Chains of Distributions Hill's Super Distribution The Scale Invariance Principle Philosophical and Conceptual Observations Some General Results Density Curves and Their Leading Digits Distributions The Case of k/x Distribution Uniform Mantissa, Varied Significand, and the General Law Uniqueness of k/x Distribution Related Log Conjecture Testing Related Log Conjecture via Simulations The Lognormal Conjecture of Hill's Super 66. Non-Symmetric Related Log 67. Wide Range on the Log-Axis and Logarithmic Distribution 275 Curves 279 Behavior The Remarkable Malleability of Related Log Conjecture Hill's Super Distribution and Related Log Conjecture Scale Invariance Principle and Related Log Conjecture The Near Indestructibility of Higher Order Distributions Falling Density Curve with a Tail to the Right Falling Density Curve with a Particular Steepness Fall in Density is Well-Coordinated Between IPOT Values Synthesis 76. Dichotomy Between the Deterministic and the Random 31 3 Between the Deterministic and the Random the Fitting Random into the Deterministic The Random Flavor of Population Data The Lognormal Distribution and Benford's Law Scrutinizing Digits within Lognormal, Exponential, 81. Leading Digits and k/x 339 Inflection Point Digital Development Pattern Found in all Real-Life Random Data Digital Development Pattern Seen Only Under IPOT Partition Development Pattern More Prevalent than Benford's Law Itself Sum-Invariant Characterization of the Law (Summation Test) 363 Section 5: Benford's Law in the Physical Sciences Mother Nature Builds and Destroys with Digits 87. Quantum Mechanics,Thermodynamics, in Mind 375 and Benford's Law 377

5 Computer XX Contents 88. Chemistry, Random Linear Combinations, and Benford's Law Benford's Law and the Set of all Physical Constants MCLT as an Explanation for Single-Issue. Physical 91. Chains as an Explanation for Single-Issue Physical Phenomenon 389 Phenomenon Breaking a Rock Repeatedly into Small Pieces is Logarithmic Random Throw of Balls into Boxes Approximating the Logarithmic Logarithmic Model for Planet and Star Formations Hybrid Causes Leading to Logarithmic Convergence Mild Deviations Seen in Small Samples of Logarithmic Data Sets The Remarkable Versatility of Benford's Law 423 Section 6: Topics in Benford's Law Singularities in Exponential Growth Series Super Exponential Growth Series Higher-Order Leading Digits Digit Distributions Assuming Other Bases Chains of Distributions Revisited Chainable Distributions and Parameters Frank Benford's Averaging Scheme as a Distribution Chain Effects of Parametrical Transformations on Leading Digits Digits of the Wald, Weibull, chi-square, and Gamma Distributions Digital Patterns of the Exponential Distribution Saville Regression Measure Revisited The Scale Invariance Principle and AGD Interpretation Case Study XI: Large Sample from a Variety 111. Direct Expression of first Digit for any Number 112. Artificially Creating Nearly Perfect Logarithmic of Data Sources 501 Use 506 Data 507 Section 7: The Law of Relative Quantities The Relating Concepts of Digits, Numbers, and Quantities Benford's Law in its Purest Form Number System Invariance Principle Cartesian Coordinate System is Number-System-Invariant Physics is Number-System-Invariant Multiplicative CLT is Number-System-Invariant Greek Parable and Chains are Number-System-Invariant 529

6 Contents xxi 120. Physical Reality versus Digital Perception Patterns in Physical Data Transcend Number Systems and Digits Common Thread Going Through Multiple Physical Data Sets Casting a Repetitive Bin System to Measure Fall in Histogram Non-Expanding Bin System Measuring 125. Once-Expanding Bin System Measuring Fall in k/x Distribution 542 Fall in k/x Distribution Once-Expanding Bins for k/x Reduces to Benford when F = D Twice-Expanding Bin System Measuring Fall in k/x Distribution Twice-Expanding Bins for k/x Reduces to Benford when F = D Infinitely Expanding Bin System Measuring Fall in k/x Confirmation Matching k/x Fall with Empirical Bins on Real Data Closed Form Expression for the Limit of the Infinite Sequence Closed Form Expression for the Limit in the Flat Case F Bin Systems with F= 10 on Real Data All Yield LOGTEN(l + l/d) Bin Systems Need to Start Near Origin with Small Initial Width Actual or Degree of Compliance May Be Bin- and Base-Variant Correspondence in Data Classification Between Bin Systems and BL F = D + 1 Bin Systems on Real Data Yield LOGBASE(l+l /d) The Remarkable Malleability and Universality of Bin Schemes Higher-Order Digits Interpreted as Particular Bin Schemes Bin Development Pattern The General Scale Invariance Principle Paradoxes Explained Digits Serving as Quantities in Benford's Law Frank Benford's Prophetic Words Future Direction The Universal Law of Relative Quantities Dialogue Concerning the Two Chief Statistical Systems 628 References 637 Glossary of Frequently Used Abbreviations 643 Index 645

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